Now consider overdispersion in the Poisson model.

(a) Suppose \(Y \mid \mu \sim \mathcal{P}[\mu]\), where \(\mu=\exp \left(\beta_{0}+\beta_{1} x\right), \beta_{0}=\gamma_{0}+\varepsilon\), and \(\varepsilon\) is an unobserved random variable with \(\mathrm{E}[\varepsilon]=0, \mathrm{~V}[\varepsilon]=\sigma^{2}>0\). Show that \(V[Y]>E[Y]\).

(b) Consider the NB2 model with the variance function \(\mu+\alpha \mu^{2}\) and the probability mass function given in (20.12). Using graphs for four different values of \(\alpha \in[0,3]\), describe the behavior of the probability mass for different realized values of \(Y\); in your answer concentrate on the behavior of the function near the origin and in the right tail.

(c) For the NB2 density given in (20.12) in Section 20.4.1, show that as \(\alpha \rightarrow 0\) the density goes to the Poisson. [This could be tricky.]

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20.4.1. Negative Binomial Model The negative binomial model, a specific example of a continuous mixture model, can be obtained in many different ways. The following justification using a mixture distri- bution is one of the oldest and has wide appeal. Suppose the distribution of a random count y is Poisson, conditional on the pa- rameter , so that f(y|2) = exp(-2)2/y!. Suppose now that the parameter is random, rather than being a completely deterministic function of regressors x. In particular, let v, where is a deterministic function of x, for example exp(x'), and v > 0 is iid with density g(v|). This is an example of unobserved heterogene- ity, as different observations may have different (heterogeneity) but part of this difference is due to a random (unobserved) component v. Note that E[2]] = if E[v] 1, so the interpretation of the slope parameters stays as in the Poisson model. The marginal density of y, unconditional on the random parameter v but conditional on the deterministic parameters and a, is obtained by integrating out v. This yields |hy| a) = (y|, v)g(va)dv, (20.11) where g(va) is called the mixing distribution and a denotes the unknown parameter of the mixing distribution. The integration defines an "average" distribution. For some specific choices of f() and g(), the integral will have an explicit or closed-form solution.